The sun and its ellipses.

Gravity falls off as the square of the distance, and that alone makes the orbits conics. From one position and one velocity the whole ellipse follows in closed form; the integrators are then held to it, and to Kepler's laws: equal areas in equal times, and T² proportional to a³. Over the centre of its ellipse a planet is exactly a from the sun, by Pythagoras.